Choice of Thresholds. for Wavelet Shrinkage Estimate of the Spectrum. Hong-ye Gao. MathSoft, Inc Westlake Ave. N., Suite 500.

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1 Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum Hong-ye Gao Statistical Sciences Division MathSoft, Inc Westlake Ave. N., Suite 500 Seattle, WA Abstract We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coecients. We propose the use of thresholds tj;n depending on sample size n, wavelet basis and resolution level j. At ne resolution levels (j = 1; 2; :::), we propose tj;n = j log n; where fjg are level-dependent constants and at coarse levels (j 1) tj;n = p 3 p log n: The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties. Key Words and Phrases: Log Spectrum Estimation; Orthogonal Wavelet Transformation; Shrinkage Estimator. Acknowledgements: This work was supported by NASA contract NCA2-488, by NSF DMS and by NSF DMS This is part of the author's Ph.D dissertation, under the supervision of Professor David L. Donoho, whose generous guidance and encouragement are gratefully acknowledged and appreciated.

2 1 Introduction Suppose we want to study a time series from an observed segment of the series X 1 ; :::; X N. There are two aspects to the study of time series analysis and modelling. The aim of analysis is to understand the nature of the process. The main reason for modelling a time series is to enable forecasts of future values of the process. No forecasting can be made before we understand the salient features of the process being characterized. The analysis of a time series can be done either in the time domain or in the frequency domain. In the time domain, attention is focused on the relationship between observations at dierent points in time, e.g. ARMA models; while in the frequency domain it is cyclical movements which are studied and this can be done by studying the spectral density h. The two forms of analysis are complementary rather than competitive. They give dierent insights into the nature of the process. Our goal in this paper is to estimate h from the data X 1 ; :::; X N. Here h(!) = 1 2 1X s=01 (s) cos(s!): The techniques currently used in spectral analysis are: Window methods (also known as kernel methods), x6.2.3 of Priestley (1981). A spectral window (also called weight function) is used to smooth the periodogram, or equivalently a lag window (also known as weight sequences, Priestley (1981), pp. 437, covariance window, Bentkus and Susinskas (1982)) W (1) is applied to the sample auto-covariances. The resulting estimate is: ^h(!) = 1 2 X jsj<n W (s)^(s) cos(s!) Typical choices of windows include the Fejer window: W (s) = I [jsjm ] for some 1 M N, for example M = p N; Bartlett window: W (s) = (1 0 jsj=m) + ; Daniell window, W (s) = sin(s=m)=(s=m); see x6.2.3 of Priestley (1981) for more choices of windows. Bentkus and Susinskas (1982) has studied optimal L 2 convergence rate for the window method over certain classes of smooth spectra. Autoregressive spectral estimation (AR approximation), Parzen (1974). A high order autoregressive model y t = 1 y t p y t0p + " t is t and the corresponding spectrum, a rational function, ^h(!) = ^ 2 2j1 0 ^ 1 z ^ p z p j 2 is taken as an approximation to f. 2

3 Mixtures of the above, e.g. the prewhitening technique, x7.4.1 of Priestley (1981). Maximum entropy (ME) methods, introduced by Burg (1967, 1972). This method is essentially equivalent to AR method. ARMA approximation: similar idea as AR approximation. They perform well in many cases and have been used frequently in practice. They have both advantages and disadvantages. For example, window methods have computational advantages but they may perform poorly in cases of high dynamic range; AR approximation is suitable only for relatively smooth spectra, Tukey (1978); ARMA procedures are almost always eective and furnishes attractive interpretations for the derived model; but it is computationally unrealistic to t high-order ARMA models, and rigorous theory of asymptotic properties is lacking. In this paper, we will develop a technique, based on wavelet decomposition of the periodogram and reconstruction of the spectrum; our new approach avoids shortcomings of earlier methods. Our new approach is computationally ecient; it can estimate spectra which are nonsmooth at a near-optimal rate; and it can be rigorously analyzed. It is well known that the asymptotic variances of the empirical wavelet coecients are functions of the unknown spectrum, and that non-linear Wavelet Shrinkage procedure below heavily depends on the asymptotic variances. Since the asymptotic variances are \proportional" to the coecients, i.e., the bigger the coecients at location (j; k), the higher the noise level at this location. This makes estimation more dicult in practice. In order to overcome the heteroscedasticity, we therefore consider estimating the log-spectrum. Suppose we observe a segment X 0 ; X 1 ; :::; X 2n01 of a Gaussian time series with mean zero and spectral density h, where for simplicity we assume that n = 2 m is dyadic. In this paper we discuss the following 4-step wavelet shrinkage procedure for estimation of log spectrum, g = log h: [1] Calculate the log-periodogram z l = log I(! l ); l = 0; : : : ; n 0 1; where! l = l=2n and X I(!) = 1 2n01 4n j X t e 0it! j 2 (1) t=0 [2] Take a standard periodic wavelet transform of (z l ) to get the empirical wavelet coef- cients fy j;k g j=1;2;:::m01;k=0;:::;n=2j 01. [3] Apply the soft threshold t (x) = sgn(x)(jxj 0 t) + (2) 3

4 to the empirical wavelet coecients fy j;k g, with level-dependent thresholds t = t j;n according to formula t j;n = p 3 p log n _ j log n (3) where ( 1 ; :::; 10 ) = (1:29; 1:09; 0:92; 0:77; 0:65; 0:54; 0:46; 0:39; 0:32; 0:27) for the commonly used compactly supported orthogonal wavelet bases (coiets, daublets and symmlets, Chapters 6 and 8 of Daubechies, 1992) and some selected n. [4] Invert the wavelet transform, producing an estimate (^g l 3 ) of the log-spectrum at the Fourier frequencies! l. This procedure falls in the general category of wavelet shrinkage estimates for noisy data. Donoho and Johnstone (1992a,b,c), have discussed the application of such thresholded wavelet transforms for recovering curves from noisy data, where the noise is assumed to have a Gaussian distribution, and they have proposed the level-independent threshold t n = p 2 log(n), where is the noise level. Such methods have also been developed in density estimation: Johnstone, Kerkyacharian and Picard (1992) propose the use of level-dependent thresholds p j. In general such wavelet shrinkage methods have a number of theoretical advantages, including near-optimal mean-squared error and near-ideal spatial adaptation. Recently, the author and others have studied the possibility of using wavelet shrinkage to de-noise periodogram data. Application of wavelet shrinkage to the log-periodogram is particularly attractive, Moulin (1993a, b), since the logarithm is the variance stabilizing transformation for the periodogram, Wahba (1980). It is easy to apply wavelet shrinkage software designed for Gaussian noise to the log periodogram, and in some cases acceptable reconstructions have been obtained. However, this will not always be the case. The noise in the wavelet coecients of the log-periodogram has a non-gaussian character; it reaches high levels somewhat more frequently than Gaussian theory would predict. Consequently, thresholds set based on Gaussian theory will not be high enough to completely suppress noise in the coecients. The thresholding we propose is based on a careful analysis of the non-gaussian character, and is somewhat larger than Gaussian theory would predict. As an example of the dierence between our method and other proposals, we present in Figure 1 a display for an AR(24) time series: (a) its log spectrum; (b) log periodogram; and wavelet reconstructions based on (c) our proposal and (d) based on Gaussian theory. It is visually evident that the spectral estimate based on Gaussian theory has spurious noise spikes, which are unrelated to the true underlying spectrum. More plots for this example, along with other examples, for dierent sample sizes and dierent wavelet bases, are given at the end of the paper. The pattern visually evident in this example is conrmed by two theorems which we prove in this paper. 4

5 Theorem 1 Under the Wahba approximation for log periodograms (see section 2.2 below), as n! 1, P f [ j [sup jy j;k 0 Ey j;k j > t j;n ]g! 0: (4) k In words, the thresholds are set high enough so that the noise does not surpass them. Theorem 2 Under the Wahba approximation for log periodograms and a compactly supported wavelet basis, at the nest level j =1, as n! 1, Similar results hold at levels j =2; 3; :::. p P fsup jy 1;k 0 Ey 1;k j > 2 log ng! 1: (5) k In words: noise spikes are almost surely to be absent for our proposal and almost surely to be present in a proposal based on Gaussian noise. The signicance of these noise spikes and the need to de-noise is more than cosmetic. In a recent article, Donoho, Johnstone, Kerkyacharian and Picard (1992) study the Gaussian white noise model, and show that by exploiting a noise-free property like (4) one can show that the estimator attains near-optimal reconstruction in a wide variety of norms simultaneously over a wide variety of smoothness assumptions. Their arguments are general and abstract and transfer to the present setting after one has established the de-noising property (4) If noise-free property like (4) is not maintained, then an inspection of their arguments will show that the resulting estimate does not achieve near optimal reconstruction in norms which measure smoothness of the reconstruction, although it may well be true that the error in `2 norm, which does not measure smoothness, is still acceptable. 2 The Choice of Thresholds Our choice of the thresholds (t j;n ) depends on the tail behavior of the empirical coecients fy j;k g. Let p j (t) = max P fjy j;k 0 Ey j;k j > tg (6) k be the tail probability of y j;k, where wavelet coecient y j;k is a standardized linear combination of the log periodogram ordinates. Then we aim to establish that n p j (t j;n )! 0; n! 1; (7) uniformly for j = 1; 2; :::; m 0 1 as this implies (4) in Theorem 1 (see (15). In this section, we would like to give two heuristic arguments, upon which our proposed thresholds (3) are based. 5

6 2.1 Normal Approximation For normally distributed wavelet coecients where " j;k N(0; 2 ) are iid, the threshold w j;k = j;k + " j;k t j;n t n = p 2 log n enjoys a variety of nice theoretical properties, Donoho and Johnstone (1992a, b, c). Under certain regularity conditions, for j close to m = [log 2 n] (technically j! 1 as m! 1), as n! 1, individual empirical wavelet coecients will be asymptotically normal: w j;k 0 Ew j;k N(0; 2 =6) this follows by applying results of Taniguchi (1979, 1980). Moulin (1993a, b) makes similar argument. So when the sample size n is large, for coarse levels (j close to m), y j;0 ; :::; y j;2j 01 are approximately normally distributed with the same variance 2 =6. One may argue that p j ( qlog(n)=3) 1 + o(1) n p ; m > j! 1: (8) 2 log n Therefore (7) holds and this leads to the rst part of our thresholds. Here \regularity" refers to the length of memory and the asymptotic normality for the empirical coecients not only depends on n (large) and j (large), but also the regularity of the time series. In general, the longer the memory, the weaker the asymptotic normality. 2.2 Wahba Approximation For small j, the normal approximation deteriorates. Under a compactly supported wavelet basis (chapters 6 and 8 of Daubechies, 1992), for xed j, the coecients of the nest level y j;k 's are linear combinations of xed number of flog I(! l )g. For example, with the Haar wavelet, y 1;k is just the dierence of two adjacent log I(! l )'s. Consider the following non-gaussian additive noise model: z l = g l + " l (9) l = 1; 2; :::; n 0 1, where (g l ) is the object to be estimated (e.g. log spectrum at frequency! l ) and " i = log( i =2) + (10) where f i ; i = 1; 2; :::; n 0 1g are independently 2 2 = exp(1=2) distributed and = 0:57721 is the Euler-Mascheroni constant. It can be shown that E" i = 0 and var(" i ) = 2 =6. Wahba (1980) proposed this as a model for the log periodogram, i.e., z l log I(! l ) and 6

7 g l log f(! l ) 0. Note that I(! 0 ) 2 1. Since the inuence of this term is negligible for n large, we will ignore this term in our discussion. For circulant time series, see x4.3 of Harvey (1989), log I(! l ) follows model (9){(10) exactly. For general stationary processes, the above model is only asymptotically true (see Theorem of Brillinger, 1981). The exact distribution of log periodogram can be found in Wittwer (1986). Now we will show that under the model (9){(10), our advertised thresholds are indeed adequate for the ner levels. Note that under model (9){(10), each y j;k is a standardized linear combination of z i 's. Let us use the very nest level, j = 1, as an example. Similar results hold for levels j = 2; 3; :::. Suppose a compactly supported wavelet basis with L coecients in the dilation equation. For example, for the Haar wavelet, L=2; D4 wavelet, L=4, etc., more examples can be found in Chapters 6 and 8 of Daubechies (1992). Then y 1;k = LX l=1 a l z l+2k0s where P s is some shift parameter for computational purpose (e.g. s = L=2) and P a l = 0; a 2 l = 1. Inequalities below leads to the threshold (3). Let a = max 1lL ja l j, from the proofs of the Theorems (see (16) and (22) of Appendix), we can show that for t large, Hence, for n large, there is a constant A so that 0:25e 02t=a p 1 (t) (et=al) L e 0t=a : (11) 0:25 n 2=a p A(log n)l 1( log n) n =a : (12) Hence thresholds at the nest level must have the form: log n, for some constant > a, and then np 1 ( log(2n))! 0; n! 1: (13) Note that the wavelet lters obey P L l=1 a 2 l = 1, which implies a < 1. Hence, for the nest level, taking = 1 always guarantees (13) and (7) follows for our proposal t j;n. In general, there are L j non-zero constants a i;j so that y j;k = L X j i=1 a i;j z i8k2 j (14) where 8 in subscript is interpreted modulo n. Let a j = max i ja i;j j, then we have L j 2 j (L 0 1) 0 L + 2 and a j 2 0(j01)=2 ; where L is the length of the lter and = sup j1 2 (j01)=2 a j < 1:534. In Tables 1 and 2 of Appendix, we list some L j 's and 2 (j01)=2 a j 's for the commonly used wavelets (apparently they converge to sup x j (x)j). This leads to the threshold (3). 7

8 3 Numerical Examples In this section, we study four time series and compare our proposed method with the one based on Gaussian theory. Figure 1 contains an AR(24) signal, with coecients: Figure 2 contains a white noise siganl, the true log spectral density in this case is g(!) = log( 2 =2), a constant. Figure 3 contains an MA(15001) time series with coecients a 1 = =4; AR method is compared in this example. a n+1 = sin(n=2) ; n = 1; 2; :::; 15000: n Figure 4 contains the Sunspots signal, WaveShrink estimate with comparisons with an AR estimate from SPLUS function spec.ar. An AR(2) model, suggested by Priestley (1981), pp. 882, is also plotted. 8

9 (a) (b) (c) (d) Figure 1: (a). log spectrum of an AR(24) time series. (b). log periodogram, (c). WaveShrink estimate based the new thresholding scheme, and (d). WaveShrink estimate based on Gaussian theory. 9

10 (a) (b) (c) (d) Figure 2: (a). log spectrum of a white noise signal, (b). log periodogram, (c). WaveShrink estimate based the new thresholding scheme, and (d). WaveShrink estimate based on Gaussian theory. 10

11 (a) (b) (c) (d) Figure 3: (a). log spectrum of a long MA series, (b). log periodogram, (c). WaveShrink estimate based the new thresholding scheme, and (d). AR estimate from SPLUS function spec.ar. 11

12 (a) (b) (c) (d) Figure 4: (a). sunspots data, (b). log periodogram, (c). WaveShrink estimate (solid line) and AR(2) estimate (dotted line), and (d). AR (of order 27) estimate. 12

13 4 Appendix 4.1 Tables L1 H1 L2 H2 L3 H3 L4 H4 L5 H5 L6 H6 C C C C C D D D D D D D D D D S S S S S S S Table 1: Maximum Coecients, Lj low-pass cascade lter at level j, Hj high-pass cascade lter at level j. C for Coiets; D for Daublets, Daubechies' original wavelets; S for Symmlet, Daubechies' near-symmetric wavelets. 13

14 L=2 L=4 L=6 L=8 L=10 L=12 L=14 L=16 L=18 L=20 j= j= j= j= j= j= Table 2: Number of Non-zero Coecients, L is the length of the lter and j = level. 4.2 Proofs of the Theorems Proof of Theorem 1: First of all, P f [ j [sup jy j;k 0Ey j;k j>t j;n ]g k m01 X j=1 m01 X j=1 P fsup jy j;k 0 Ey j;k j > t j;n g k 2 j p j (t j;n ) n max p j(t j;n ) (15) Under model (9) and (10), from (14), coecients in level j can be written as y j;k 0 Ey j;k = L X j i=1 a i;j " i8k2 j with P i a i;j = 0 and P i a 2 i;j = 1. Let a j = max i ja i;j j. The moment generating function for " log( 2 2=2) is where 0 is Euler's Gamma function. Let 3 3 j(t) = inf s f0st + Lj M(u) Ee u" = 0(u + 1) X i=1 (log 0(1 + a i;j s) + log 0(1 0 a i;j s))g It can be shown that for any random variable X with EX = 0, for all a > 0 and t. Then and therefore, P (X > a) e 0at Ee tx p j (t) e 33 j (t) max p j(t) sup e 33(t) j = exp( sup 3 3 j (t)) (16) 14

15 Let `n = (log n) 7=6 and consider t 3 j = 8 < : log n L 1=4 j p log n L j < `n L j `n : (17) For the thresholds we have t j;n t 3 j. Therefore, t j;n = log n L 1=4 j _ p log n and sup p j (t j;n ) sup p j (t 3 j) exp( sup 3 3 j (t 3 j)) _ sup 3 3 j (t3 j ) sup 3 3 j (t3 j ) sup 3 3 j (t3 j ) 1L j <`n `nl j n = sup 1L j <`n When L j < `n = (log n) 7=6 and n large, we have log n > L 1=4 j 3 3 j( log n _ p ) sup 3 3 j( log n) (18) L 1=4 `nl j j n q L j a j L j Consider the following non-positive function H, dened on R + by Lemma 1 For t a j L j, and H(t) = 0t log t 3 3 t j(t) L j H( ) a j L j L H( log n ) # in L for large n and 1 L `n: al3=4 The proof will be given later. Combining (16) and Lemma 1 yields the second part of (11). From the Lemma, sup 1L j <`n 3 3 j( log n ) sup L 1=4 1L j j <`n L j H( log n ) al 3=4 j H( a log n) = 0 log n (1 + o(1)) (19) a and When L j `n, we have p log n L 3=7 j 15

16 Lemma 2 For j 2 (0; 1) and 0 t 2 j =6a j, t j(t) 0 : 2 =3 + 6 j In particular, for n large and j 6a j L 3=7 j = 2 L 01=14 j (log n) 01=12, 3 3 j The proof will be given later. From the Lemma, (t) 03t2 (1 + o(1)) t 2 L3=7 j : p sup 3 3 j( log n) 0 32 log n (1 + o(1)): (20) `nl j n 2 Combining (18), (19) and (20), we have sup 3 3 j(t 3 j ) 0( a Therefore, for any > a and > = p 3, n sup and this completes the proof of Theorem 1. ^ 3 ) log n (1 + o(1)): 2 p j (t j;n ) n exp( sup 3 3 j (t j;n )) = o(1): Proof of Lemma 1: First of all, it can be shown that for 0 x < 1, 0(1 + x) 0(1 0 x) and 0(x) 1=x. (In fact, 0(x) = Q x 01 1 (1+1=k) x k=1 1+x=k and (1 + 1=k) x 1 + x=k when 0 x < 1.) Therefore, 3 3 j (t) inf jsj<1=a j f0st + L X j i=1 log 0(1 + a i;j s)g L j inf f0 st 0 log(1 0 s)g 0s<1 a j L j and the rst part of the Lemma can be easily solved from the last expression. For n large, such that C n a 01 log n > 11 and `n = (log n) 7=6 (C n =11) 4=3. Let h(x) = xh(c n x 03=4 ), then for 1 x < `n, h 0 (x) is increasing in x and h 0 (`n) < 0. Therefore, h 0 (x) < 0 for 1 x < `n and this means h(x) is decreasing in x. The proof is completed. Proof of Lemma 2: Let K(x) = log 0(1 + x), then for x > 01, by Taylor expansion, K(x) = K(0) + K 0 (0)x + x 2 K 00 (x)=2 = 0x + x 2 K 00 (x)=2; (21) 16

17 where = 0:5772:::: is the Euler constant and = (x) 2 (0; 1). Note that K 0 (x 0 1) is the famous Psi function. By the properties of Psi function (Davis (1935), pp. 11), K 00 (x) = P 1 k=1 (k + x) 02 and K 00 (0) = 2 =6. Hence, for any 2 (0; 1), we have K 00 (0) K 00 (0) + 3 K and for x 0, Recall that P i a i;j = 0 and P i a 2 i;j = 1, K(x) 0x + x 2 K =2: X 3 3 j (t) = inf f0st + s i K(a i;j s)g and Lemma 2 follows immediately. X inf f0st + (0a i;j s + a 2 i;j jsj< j =a s2 K =2)g j i = inf f0st + s 2 K =2g jsj< j =a j Proof of Theorem 2: To prove Theorem 2, we need a lower bound for p j (t). It is easy to show that if X; Y are independent, then for any s; t > 0, P fjx + Y j tg P fjxj < sgp fjy j t + sg: Suppose ja 1 j = max ja l j = a. There exists t 0 > 0 such that for t t 0, P fj LX l=2 a l log( l =2)j < tg 1 2 : Also notice that for 0 x 1, 1 0 e 0x x=2. Then p 1 (t) = P fj LX l=1 a l log( l =2)j tg P fja 1 log( 1 =2)j 2tgP fj 1 2 P fj log( 1=2)j 2t a g 1 2 P flog( 1=2) 0 2t a g and this completes the proof of Lemma 1. LX l=2 a l log( l =2)j < tg = 1 2 f1 0 exp(0e02t=a )g 1 4 e02t=a : (22) From (12) we can see that for any constant C < a=2, np 1 (C log n)! 1; n! 1: 17

18 In particular, p s 2 np 1 ( 2 log n) = np 1 ( log n)! 1; n! 1: log n There exists K > 0 such that fy 1;iK g are independent (linear combinations over disjoint intervals). Therefore, P fsup k p p jy 1;k 0 Ey 1;k j > 2 log ng P fsup jy 1;iK 0 Ey 1;iK j > 2 log ng i Y and this implies the Theorem 2. n=2k = 1 0 i=0 (1 0 p 1 ( p 2 log n))! 1; References [1] Bentkus, R. Yu. and Susinskas, Ju. V. (1982) On Optimal Statistical Estimators of the Spectral Density. Soviet Math. Dokl., Vol. 25, No. 2, [2] Brillinger, D. R. (1981). Time Series: Data Analysis and Theory, Holden-Day, San Francisco. [3] Burg, J. P. (1967). Maximum Entropy Spectral Analysis. Paper Presented at the 37th Annual international S.E.G. Meeting, Oklahoma City, Oklahoma. [4] Burg, J. P. (1972). The Relationship between Maximum Entropy Spectra and Maximum Likelihood Spectra. Geophysice, 37, [5] Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM. Philadelphia. [6] Davis, H. T. (1935). Tables of The Mathematical Functions, Vol II, Trinity University. [7] Donoho, D. (1992). De-Noising By Soft-Thresholding. Technical Report, Department of Statistics, Stanford University. [8] Donoho, D. and Johnstone, I (1992a). Minimax Estimation via Wavelet Shrinkage. Technical Report, Department of Statistics, Stanford University. [9] Donoho, D. and Johnstone, I (1992b). Adapting to Unknown Smoothness via Wavelet Shrinkage. Technical Report, Department of Statistics, Stanford University. [10] Donoho, D. and Johnstone, I (1992c). Ideal Spatial Adaptation by Wavelet Shrinkage. Technical Report, Department of Statistics, Stanford University. [11] Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1992). Wavelet Shrinkage: Asymptopia? Technical Report, Department of Statistics, Stanford University. 18

19 [12] Gao, Hong-Ye (1992). Spectral Density Estimation by Using Wavelets. Technical Report, Department of Statistics, Stanford University. [13] Harvey, A. C. (1981). Time Series Models, Philip Allan. [14] Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press. [15] Johnstone, I., Kerkyacharian, G. and Picard, D. (1992). Estimation d'une densite de probabilite par methode d'ondelettes. Comptes Rendus Acad. Sciences Paris (A), [16] Moulin, P. (1993a) A Wavelet Regularization Method for Diuse Radar-Target Imaging and Speckle-Noise Reduction. Journal of Math. Imaging and Vision, Special Issue on Wavelet, Vol. 3, [17] Moulin, P. (1993b) Wavelet Thresholding Techniques for Power Spectrum Estimation. Submitted to IEEE Trans. on Signal Processing. [18] Parzen, E. (1974). Some Recent Advances in Time Series Modelling. IEEE Trans. Automatic Control, AC-19, [19] Priestley, M. B. (1981). Spectral Analysis and Time Series, Academic Press, London. [20] Taniguchi, M. (1979). On Estimation of Parameters of Gaussian Stationary Processes. J. Appl. Probab., 16, [21] Taniguchi, M. (1980). On Estimation of the Integrals of Certain Functions of Spectral Density. J. Appl. Probab., 17, [22] Tukey, J. W. (1978). When Should Which Spectrum Approach Be Used? Paper Presented at the Institute of Statisticians Conference, King's College, Cambridge, July, [23] Wahba, G. (1980). Automatic Smoothing of the Log Periodogram. Journal of the American Statistical Association, 75, [24] Wittwer, G. (1986) On the Distribution of the Periodogram for Stationary Random Sequences (with Discussion). Math. Opera. und Stat., 17,